On melting for the 3D radial Stefan problem

We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \begin{equation*} \left\{\begin{aligned} &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad in\ r\geq\lambda(t),\\ &\partial_{r}u(t,\lambda(t))=-\dot{\lambda}(t),\\ &u(t,\lambda(t))=0,\\ &u(0,\cdot)=u_{0},\quad \lambda(0)=\lambda_{0}. \end{aligned}\right. \end{equation*} We prove the existence in the radial class of finite time melting with rates \begin{equation*} \lambda(t)=\left\{\begin{aligned} &4\sqrt{\pi}\frac{\sqrt{T-t}}{|\log (T-t)|}(1+o_{t\rightarrow T}(1)),\\ &c(u_{0},k)(1+o_{t\rightarrow T}(1))(T-t)^{\frac{k+1}{2}},\quad k\in{\mathbb{N}}^{*}, \end{aligned}\right. \end{equation*} which respectively correspond to the fundamental stable melting rate and a sequence of codimension $k$ unstable rates. Our analysis mainly depend on the methods developed in [17] which deals with the similar problems in two dimensions and also the construction of both stable and unstable finite time blow-up solutions for the harmonic heat flow in [49],[50].